Question

Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Develop the least squares estimated regression equation. What is the coefficient of determination? x y 2 12 3 9 6 8 7 7 8 6 7 5 9 2

Answer 1

Answer:

Step-by-step explanation:

Hello!

Given the independent variable X and the dependent variable Y (see data in attachment)

The regression equation is

^Y= b₀ + bX

Where

b₀= estimation of the y-intercept

b= estimation of the slope

The formulas to manually calculate both estimations are:

$b= \frac{sumXY-\frac{(sumX)(sumY)}{n} }{sumX^2-\frac{(sumX)^2}{n} }$

$b_0= \frac{}{y} - b*\frac{}{x}$

n=7

∑X= 42

∑X²= 292

∑Y= 49

∑Y²= 403

∑XY= 249

$\frac{}{y} = \frac{sumY}{n} = \frac{49}{7} = 7$

$\frac{}{x} = \frac{sumX}{n} = \frac{42}{7} = 6$

$b= \frac{249-\frac{42*49}{7} }{292-\frac{42^2}{7} }= -1.13$

$b_0= 7- (-1.13)*6= 13.75$

^Y= 13.75 - 1.13X

Using the raw data you can calculate the coefficient of determination as:

$R^2= \frac{b^2[sumX^2-\frac{(sumX)^2}{n} ]}{[sumY^2-\frac{(sumY)^2}{n} ]}$

$R^2= \frac{(-1.13)^2[292-\frac{(42)^2}{7} ]}{[403-\frac{(49)^2}{7} ]}= 0.84$

This means that 84% of the variability of the dependent variable Y is explained by the response variable X under the model ^Y= 13.75 - 1.13X

I hope this helps!